Non Linear Contracting and Endogenous Buyer Power between Manufacturers and Retailers: Empirical Evidence on Food Retailing in France

the first derivatives of the J retail prices with respect to the J' wholesale prices.

Remark that the last J — J' rows of this matrix are zero because they correspond to private label
products for which wholesale prices have no meaning.

Then, we can write the first order conditions (3) in matrix form and the vector of manufacturer’s

• 2
margins is

Γ ≡ w — μ = —(If P_wS_pIf ) ¹If s(p~)

The first derivatives of retail prices with respect to wholesale prices depend on the strategic inter-
actions between manufacturers and retailers.

When we assume that retailers follow manufacturers in setting the retail prices given the who-
lesale prices, P_w can be deduced from the differentiation of the retailer’s first order conditions (1)
with respect to wholesale price, i.e. for j ∈ S_r and к = 1,.., J'

⅛s _l ^d^+У ^d^ '+У (p, —w,—d) У ^d⅛^   =0 (5)

^{{      d}Pj   ⅛ ^dPj ^dwk  “              , ^dPj ∂p_s ∂w_k

I 1<У .                           I ^z *У .                       S 1 ^_aa^,

where 1 ; _k. .s', . =1 if к ∈ S_r and 0 otherwise. Defining S_p^p the matrix of the second derivatives of

the market shares with respect to retail prices whose elements are :
we can write equation (5) in matrix form³ :

IrPw = (Ir — Ir)spIr [SpIr + IrspIr + (SpⁱIr7∣...∣sp^jIr7)I_r] ¹               (6)
where 7 = p — w — c, I_r is the ownership matrix of private labels of retailer r and I_r — I_r thus

designates the ownership matrix of national brands by retailer r. Equation (4) shows that one can

²Rows of this vector that correspond to private labels are zero.

³We use the notation (α∣b) for horizontal concatenation of a and b. The full matrix P_w can be obtained by
summing over r these expressions.

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